Maybe it's just a matter of interpretation, but...
- My nephew = o meu sobrinho
A nephew of mine = Um sobrinho meu
His nephew = o seu sobrinho/o sobrinho dele
- A nephew of his = um sobrinho seu/um sobrinho dele (As Lahure said, it isn't an apostrophe, but I think the meaning is the same.)
Ok, there are two Brazlian songs that don't respect that. Maybe that has something to do with eyes...
Quando a luz dos olhos meus e a luz dos olhos teus resolvem se encontrar.
Olhando assim nos olhos teus, sei que vai ficar nos meus a marca desse olhar.
Anyway, I think the article makes a difference to interpret dele.
No, I meant that, in Portuguese, the possessive generally appears after the noun if it's an indefinite noun, as our National Anthem says "Verás que um filho teu não foge à luta". So "o sobrinho dele", of course, it is "his nephew"; but "um sobrinho dele" can be understood as "a nephew of his" regardless of being standard or not. Anyway, I don't consider "um sobrinho dele" as apostrophe, because this would actually be "um sobrinho seu" in third person, but the meaning is the same.
There's actually a simple reason for this — by default, a prepositioned possessive without article works as a determiner rather than a mere adjective, and therefore "implies" definitiveness (similar to English, which actually rejects coapparition of articles and possessive adjectives because both are always determiners). To force the indefinitive interpretation in Portuguese, one most include an indefinite article (and then the possessive becomes a "mere" adjective).
Hence meu sobrinho / sobrinho meu = my nephew o meu sobrinho / o sobrinho meu = my nephew / the nephew of mine um meu sobrinho / um sobrinho meu = one/a nephew of mine
Obviously, there is a strong preference in speech for prepositioning the possessive when it's definitive, and postpositioning it when it's indefinite, although both variations are perfectly valid Portuguese.